Highly non-classical states of light - such as the approximate Gottesman-Kitaev-Preskill states or cat-like states - can be generated from experimentally accessible Gaussian states via photon counting measurements on selected modes, conditioned on specific outcomes of these heralding events. A simplest yet important example of this approach involves performing photon number measurements on one mode of a two-mode entangled Gaussian state. The heralding probability of this scheme is a key figure of merit, as it determines the generation rate of the targeted non-classical state. In this work we show that the maximum heralding probability for the two-mode setting can be calculated analytically, and we investigate its dependence on the number of detected photons n. Our results show that the number of required experimental trials scales only polynomially with n. Generation of highly complex optical quantum states with high stellar rank is thus practically feasible in this setting, given access to sufficiently strong squeezing.
Jaromı́r Fiurášek (Thu,) studied this question.